Application of fuzzy logic to improve the Likert scale to measure latent variables

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The research studied the process of improving the Likert scale based on fuzzy logic to measure latent variables and to compare the quality of the data as measured by the improved Likert scale with data measured by the Likert scale. Qualitative study and survey study were used as the research methodology.

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Kasetsart Journal of Social Sciences 38 (2017) 337e344 Contents lists available at ScienceDirect Kasetsart Journal of Social Sciences journal homepage: http://www.elsevier.com/locate/kjss Application of fuzzy logic to improve the Likert scale to measure latent variables Paothai Vonglao Faculty of Science, Ubon Ratchathani Rajabhat University, Ubon Ratchathani 34000, Thailand a r t i c l e i n f o a b s t r a c t Article history: The research studied the process of improving the Likert scale based on fuzzy logic to Received 16 February 2016 measure latent variables and to compare the quality of the data as measured by the Received in revised form 21 December 2016 improved Likert scale with data measured by the Likert scale. Qualitative study and survey Accepted 30 January 2017 study were used as the research methodology. Data analysis included content analysis and Available online 26 August 2017 statistics comprising the arithmetic mean, standard deviation, standard error, consensus index, and the KolmogoroveSmirnov test. It was found that the Likert scale could be Keywords: improved by using Mamdadi fuzzy inference which included four important steps: fuzzy logic, (1) fuzziﬁcation, (2) fuzzy rule evaluation, (3) aggregation, and (4) defuzziﬁcation. A latent variable, comparison of the two different approaches showed that the data measured using the Likert scale improved Likert scale was more suitable to be analyzed with the arithmetic mean and standard deviation than the data measured using the Likert scale. More importantly, the distribution of data measured by the improved Likert scale was normal with a lower standard error, making it appropriate for data analysis for statistical inference. © 2017 Kasetsart University. Publishing services by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/ 4.0/). Introduction interval scale as they are acquired through psychological scaling. The latent variables are measured by the com- Internal validity of quantitative research is a measured bined scores of all questions, which are on an interval scale validity. Thus, the instrument which is used to collect data (Tirakanan, 2008, p. 57). However, many scholars have on the variables measured is important. Subjective vari- argued that naturally, in the Likert scale, the choice or ables are latent traitsdthey are not directly observable or answer is only the data organized on an ordinal scale measurable. Instead, they are measurable through feel- (Hodge & Gillespie, 2003; Pett, 1997). With reference to ings, behaviors, expressions, and personal opinions, and the Likert scale, Cohen, Manion, and Morrison (2000) data can be acquired using a questionnaire. The Likert stated that the interval range of different levels are not scale is one of the popular instruments to measure such equal in value. The Likert scale, thus, should be arranged latent traits. The scale was introduced by Likert (1932) and on an ordinal level. It is inappropriate to analyze the data consists of a series of questions which are indicators of the using addition, subtraction, division, or multiplication. latent traits. Each question has a ﬁve-scale response: least, Furthermore, it is inappropriate to analyze such data using less, moderate, more, and most with the scores for the the arithmetic mean and standard deviation (Clegg, 1998). scale being 1, 2, 3, 4, and 5, respectively. Edward (1957) Thus, it is inappropriate to measure the latent variables by stated that the scores in question are based on an combining the scores of all the items from a Likert scale. In addition, Sukasem and Prasitratsin (2007, p. 2) explained that researchers in general would combine the scores from E-mail address: paothai@hotmail.com. each item, and then use the combined scores to measure Peer review under responsibility of Kasetsart University. http://dx.doi.org/10.1016/j.kjss.2017.01.002 2452-3151/© 2017 Kasetsart University. Publishing services by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http:// creativecommons.org/licenses/by-nc-nd/4.0/).
338 P. Vonglao / Kasetsart Journal of Social Sciences 38 (2017) 337e344 the variables, which is incorrect as each item is unequal in doing this, it is possible to avoid the answer of ‘moder- its weight. ate’. Hodge and Gillespie (2003) proposed that the Because of the problems described above, many at- question should be divided into two parts. First, the tempts have been made to deal with this issue and to leading question was raised to encourage respondents to develop a suitable scale. One of the methods is fuzzy logic. express their feelings, which was followed by a secondary It was developed from a fuzzy set by Zadeh (1965). Lalla, question on the contents of the leading questions, both Facchinett, and Mastroleo (2004) and Li (2013) applied positive and negative. The respondents can choose from fuzzy logic to improve the Likert scale, which resulted in a 0 to 10 depending on the intensity. However, this method new scale known as the fuzzy Likert scale (FL). Li also may not be effective, as the respondents can get lazy in compared the efﬁciency of this scale with the Likert scale answering all the questions. Li (2013) proposed the and found that measuring the variables using the fuzzy construction of the fuzzy Likert scale (FLS). The re- Likert scale was more accurate than measuring with the spondents have only one choice. Its membership value general Likert scale. For the reasons described, the current lies between 0 and 1. That is, if an opinion is inclined research tried to determine the process for applying fuzzy towards that choice, its value is set at 1. On the contrary, logic to the Likert scale to measure the latent variables in a if the opposite happens, the answer is an ordered pair. more valid and efﬁcient manner. It is expected that the The ﬁrst is an answer and the second is the value of research would lead to measuring methods which are more membership. The acquired answer is adjusted into the effective and appropriate. fuzzy Likert scale: P Literature Review uo Ao FLS ¼ P (1) Ao Attitude Measuring Using the Likert Scale where, FLS is the fuzzy Likert scale. uo is to the level of an opinion according to the Likert scale, and Ao is the area of Attitude is an important variable with latent traits. Ac- the membership function that is truncated by the mem- cording to Saiyot and Saiyot (2000, pp. 52e60) attitude bership value. Although the improved scale may provide means the emotions and feelings of a person coming from more details and greater reliability, there are disadvantages an experience in learning something called a target. From as respondents may ﬁnd it hard to decide and they may get learning, there appears a feeling of like or dislikes, agree- bored. As a consequence they may not give genuine ment or disagreement. That tendency runs from a low to a answers. high intensity. Likert (1932) was the ﬁrst to propose the method to measure an attitude by combining the scores of each question. This method was called summated rating Fuzzy Logic (Tirakanan, 2008, pp. 191e192). However, the Likert scale has a disadvantage; it is unclear whether the data Fuzzy logic originated from the dissertation of Zadeh measured are based on an ordinal level or interval level (1965). It is based on the principles that out of all things (Jamieson, 2004). Although Likert assumed the data ac- in the world, there is a small portion that is certain. Things quired were based on an interval level, it can be observed are mainly uncertain. The things which are uncertain are that the data measured by the Likert scale are based on characterized by two traits: random and fuzzy. ordinal order (Hodge & Gillespie, 2003; Pett, 1997). Data on The classical set is an undeﬁned term, as it characterizes an interval level show an equal range for two consecutive a group consisting of various members which are identiﬁ- values, whereas the feeling measured by the Likert scale able. However, there are a lot of groups which cannot be has a different interval range between two levels (Cohen explicitly identiﬁed. The group having such characteristics et al., 2000). As a result, the Likert scale cannot estimate is called a fuzzy set. It refers to the set of things for which it varying interval ranges between data (Russell & Bobko, cannot be identiﬁed whether each thing in question is a 1992). What can be measured by the Likert scale is only member of the set or not. Nevertheless, it is possible to the information which cannot distinguish the interval. indicate the tendency of something to be a member of a set Furthermore, alternative forms of the Likert scale are through the membership function whose value ranges similar. Respondents have to choose only one option, which between 0 and 1. If the membership value of something is unrealistic and unreliable (Hodge & Gillespie, 2003; gets closer to 1, that has a high level of membership. By Orvik, 1972). contrast, if the membership value gets closer to 0, it has a Consequently, due to these explained disadvantages of low level of membership. the Likert scale, it is apparent that the data acquired may Deﬁnition. If X is not an empty set, x is any member of X be unreliable. Several academics have attempted to and A is a fuzzy set whose membership function is mA, then improve the Likert scale. Chang (1994) proposed that fuzzy set A can be written in the form of a pair set as more levels of the scale should be added so that more follows: details could be obtained. However, it may be difﬁcult for respondents to identify their genuine feelings at such a A ¼ fðx; mA ðxÞÞ=x2Xg; mA ðxÞ : X/½0; 1 level of detail (Russell & Bobko, 1992). Albaum (1997) proposed two steps. First, there are only two choices: Membership function is used to determine the mem- agree or disagree. After that the respondents have to bership level for x. There are many types of membership answer according to the intensity level: less or more. By function. Which type is to be used depends on suitability
P. Vonglao / Kasetsart Journal of Social Sciences 38 (2017) 337e344 339 and relevant information based on the expert's consid- (Mamdani & Assilian, 1975). The fuzzy inference in ques- eration. The types include triangular membership func- tion consists of four stages: tion, trapezoidal membership function, Gaussian membership function, and bell-shaped membership Stage 1: Fuzziﬁcation: in this stage, experts take into account function. Each function has different parameters and details concerning input, output, and results. The shape. For example, the triangular membership function input and output are considered as input and output has parameters consisting of three values: real number a, variables. Then, deﬁned linguistic variables are used b, c, for a b c. The function value can be set as follows to explain each variable. The linguistic variables (see Figure 1). determine the fuzzy set and its membership func- 8 tion. Then, the fuzzy rules are established to show < ðx aÞ=ðb aÞ; a x b the relations between input and output. mA ðxÞ ¼ ðc xÞ=ðc bÞ; b < x c (2) Stage 2: Fuzzy rule evaluation: the membership function : 0; elsewhere value of each rule is established using Equation (3). Any given system consists of input and output. System h i experts know the relations relating to these two factors. mL ðxÞ ¼ min mAL1 ðx1 Þ; mAL2 ðx2 Þ; …; mALn ðxn Þ (3) The input is the cause and the output is the result. Both are explained in linguistic variables as follows: less, moderate If the value of the membership function of any rule is and more. The variable is explainable by a fuzzy set. To equal to zero, it will not be considered. If the value of a control the system, the experts will design the causal re- membership function is not equal to zero, the value will be lations between input and output: IF input THEN output. used to truncate or scale the shape of the output mem- This is called a fuzzy rule. The number of fuzzy rules de- bership function in this rule. pends on the number of linguistic variables used to explain input and output. A general form of the fuzzy rules can be Stage 3: Aggregation: the fuzzy set of the output in stage 2 determined as follows. is combined by a union operation. Supposing that a system has n inputs and 1 output. Stage 4: Defuzziﬁcation: the fuzzy set which results from Causal relation between the factors can be illustrated with L the combined rules in stage three is changed into rules. The input is explained with linguistic variable: Aij; a crisp value. There are several methods, one of i ¼ 1,2,3,…,L and j ¼ 1,2,3,…,n. The output is explained with which is seeking a center of gravity (COG). The the linguistic variable: Ci; i ¼ 1,2,3,…,L. Let COG of fuzzy set in the range [a,b] can be deter- x ¼ [x1,x2,x3,…,xn] be a value of the input and y be a value of mined using Equation (4). the output. A general form of ith rule of the fuzzy rule of Mamdani is: Zb mA ðxÞxdx IF ð x1 is Ai1 Þ AND ð x2 is Ai2 Þ AND … a COG ¼ (4) AND ð xn is Ain Þ THEN ðy is Ci Þ Zb mA ðxÞdx Application of Fuzzy Logic a Fuzzy logic can be applied to decide or control a system Methods through the principle of fuzzy inference. Fuzzy inference has two important methods: Mamdani fuzzy inference and Participants Sugeno fuzzy inference. In this paper, only the former is described. Mamdani fuzzy inference was ﬁrst proposed in The target population was ﬁrst year students in the 1975 by Professor Ebrahim Mamdani of London University Faculty of Science Ubon Ratchathani Rajabhat University in Figure 1 Triangular membership function with parameter a ¼ 1, b ¼ 2 and c ¼ 3
340 P. Vonglao / Kasetsart Journal of Social Sciences 38 (2017) 337e344 the 2014 academic year. The total number of the students in based on the Likert scale: strongly disagree, disagree, the study was 302 (Policy and Plan Division, 2014). neither agree nor disagree, agree, and strongly agree. All these levels cannot be categorically separated. In other Data Collection words, one level overlaps with others where there is an ambiguous opinion. In addition, the ambiguity of opinion The research instrument was a ﬁfteen-item question- depends on the quality of the question in terms of validity naire to assess attitude toward mathematics based on a and discrimination. Thus, the answer is not real. Hence, to ﬁve-point Likert scale. The format in question was adapted measure the value of latent variables, it is necessary to from the one used by Saiyot and Saiyot (2000, p. 98). Five consider sharing the validity and discrimination with the experts were asked to evaluate its validity. It was found that answer of each item. Thus, we can apply fuzzy logic to the value of index of item-objective congruence (IOC) improve the answer from the Likert scale by using Mam- ranged from 0.6 to 1. Then, the questionnaire was tried out dani inference in four stages. with 50 ﬁrst year general science students in the Faculty of Education Ubon Ratchathani Rajabhat University in the Stage 1: fuzziﬁcation: in each question, it is necessary to 2014 academic year. Items having a discrimination value of determine three inputs: opinion of respondents greater than 0.2 were selected. As a result, 12 items were (O), validity (V), and discrimination (R). The acquired. The questionnaire of 12 items was administered output is a suitable answer (T). The linguistic with the target population. Data were collected using the variables which are used to explain the opinion questionnaire from 302 students who were ﬁrst year stu- are Strongly disagree (SD), Disagree (D), Neither dents in the 2014 academic year regarding their attitude agree nor disagree (NN), Agree (A) and Strongly towards mathematics. Samples were chosen from the 302 agree (SA). Validity could be explained in terms of respondent questionnaires based on simple random sam- less (L), moderate (M), and more (G). Discrimi- pling with sample sizes of 30, 40, 50, …, 200, respectively. nation could be explained in terms of less (L), Data from each sample size were collected to compare the moderate (M), and more (G). The suitable answer quality of data in each sample size with regard to inference. can be explained in terms of least (SL), less (L), moderate (M), more (G), and most (SG). The Data Analysis membership function of the linguistic variables is shown in Figures 2e5. Fuzzy logic was applied to improve the Likert scale using content analysis. The MATLAB software was then used to In total, 29 fuzzy rules were made by the experts. Some acquire a suitable response based on the applied process. of them are given below. The quality of data which were acquired from the improved Likert scale was compared with data acquired from the Rule 1 IF (O is SD) and (V is L) and (R is L) THEN (T is SL) Likert and fuzzy Likert scales. The statistics used were Rule 2 IF (O is SD) and (V is L) and (R is L) THEN (T is L) arithmetic mean ðXÞ, standard deviation (S.D.), standard « error (S.E.) and consensus index (Cns) (Tastle & Wierman, Rule 28 IF (O is A) and (V is G) and (R is G) THEN (T is G) 2007). The consensus index can be computed using Equa- Rule 29 IF (O is SA) and (V is G) and (R is G) THEN (T is SG) tion (5). Stage 2: Fuzzy rule evaluation: the value of inputs including the opinion level, validity, and discrimination is X n jxi mx j CnsðXÞ ¼ 1 þ pi log2 1 (5) used to ﬁnd the membership value of each input i¼1 dx from each fuzzy rule. If the rule has a membership function value equal to zero, it is not considered. If where, Cns(X) is consensus; X is an opinion; xi is an opinion the value of membership function is not equal to level i; n stands is the number of the opinion level; pi ids the zero, it is used to truncate the shape of the output ratio of the sample whose opinion is at level i; dx is the membership function. difference between the maximum and minimum for an Stage 3: Aggregation: the fuzzy set of the output, which is opinion; mx is the mean of an opinion for all samples. The truncated, is combined by a union operation. index of consensus ranged from 0 to 1. If it is close to 1, it Stage 4: Defuzziﬁcation: getting a suitable answer by indicates that the opinion of the samples is in accordance converting the fuzzy set which was combined in with the issue of their interest. On the contrary, if it is close stage 3 into a crisp value through COG; the value to 0, it indicates that the opinion of the samples is con- acquired is a suitable answer for the question. It tradictory to the issue in question. is called an improved Likert scale. Results Comparison of the Quality of Data Process to Improve Likert Scale The answer for each item of the Likert scale that was By applying fuzzy logic, we assume that the latent var- improved by using the process of the prior section is shown iable is measureable by using the question about that var- in Table 1. The attitude toward mathematics as measured iable. The respondent should be asked how much he or she by the Likert scale and the improved Likert scale is shown agreed or disagreed. An opinion should have ﬁve levels in Table 2.
P. Vonglao / Kasetsart Journal of Social Sciences 38 (2017) 337e344 341 Figure 2 Bell-shaped membership function used to explain the opinion of the respondents Figure 3 Trapezoidal membership function and triangular form used to explain validity Figure 4 Trapezoidal membership function and triangular form used to explain discrimination Figure 5 Trapezoidal membership function and triangular form used to explain suitable answers
342 P. Vonglao / Kasetsart Journal of Social Sciences 38 (2017) 337e344 Table 1 Table 3 Likert scale improved by applying fuzzy logic Statistics according to distribution of data as measured by the Likert scale Item IOC Discrimination Likert scale Case Likert scale X S.D. Cns 1 2 3 4 5 1 2 3 4 5 1 0.8 0.551 1.46 2.24 3.07 4.05 4.55 1 0 0 100 0 0 3 0 1 2 1 0.401 1.33 2.22 3.06 4.05 4.68 2 50 0 0 0 50 3 2.01 .00 3 0.8 0.359 1.46 2.24 3.07 4.05 4.55 3 15 15 40 15 15 3 1.23 .58 4 1 0.576 1.40 2.23 3.07 4.05 4.61 5 1 0.369 1.40 2.23 3.07 4.05 4.61 6 1 0.651 1.39 2.23 3.07 4.05 4.62 7 1 0.356 1.43 2.24 3.07 4.05 4.58 not truly reﬂect the data. However, data as measured by the 8 0.8 0.29 1.57 2.18 3.10 3.98 4.44 improved Likert scale had a different arithmetic means in 9 0.6 0.62 1.50 2.24 3.08 4.04 4.51 10 1 0.44 1.27 2.22 3.06 4.05 4.74 all cases with 3.10, 3.01, and 3.07 respectively, showing that 11 1 0.621 1.46 2.24 3.07 4.05 4.55 the arithmetic mean could truly reﬂect the data. In case 3, 12 1 0.464 1.23 2.22 3.06 4.05 4.78 the standard deviation of data as measured by the Likert scale and the fuzzy scale was equal to 1.23, which shows that the standard deviation obtained by using the two Table 2 scales cannot reﬂect the data. However, the data as Population mean and standard deviation of attitude toward mathematics measured by using the improved Likert scale had a stan- measured by the Likert scale and the improved Likert scale dard deviation equal to 0.99, which was more Item Likert scale Improved Likert coherent(Cns ¼ 0.55) and the standard deviation used to scale analyze data could truly reﬂect the data. m s m s Table 6 shows that the data measured using the Likert scale had a distribution different from a normal distribu- 1) I study mathematics with 2.65 0.84 2.80 0.72 tion with a statistical signiﬁcance of .05. The data measured relative comfort. 2) Solving mathematical 2.80 0.89 2.91 0.78 by the improved Likert scale showed a normal distribution questions is fun. at the .05 signiﬁcance level. 3) Solving mathematical 3.27 0.97 3.34 0.82 Table 7 shows that the standard error of the sample questions is boring. mean of the data measured by the improve Likert scale was 4) Learning mathematics is 3.32 1.02 3.37 0.87 boring. less than the standard error of the sample mean of the data 5) Mathematic is basic to life. 3.54 1.09 3.53 0.89 measured by the Likert scale. 6) I like calculating without the 2.66 0.97 2.79 0.83 help of a calculator. 7) Mathematic knowledge is 3.33 0.97 3.38 0.81 Discussion fundamental to all subjects. 8) Mathematics is most 3.35 1.01 3.35 0.79 The improved Likert scale with fuzzy logic was more valuable. effective than the Likert scale and the fuzzy Likert scale 9) I turn my face away when I 3.37 1.00 3.41 0.82 see mathematics books. because its scale is continuous. In addition, the mean and 10) I like to think about or 2.74 0.84 2.86 0.75 standard deviation reﬂect the fact that the data were reﬂect on mathematics. measured using the improved Likert scale. In particular, the 11) Mathematics is a terrible 3.46 1.00 3.48 0.82 standard deviation of the data is in accord with the subject. 12) The majority of people do 2.68 1.12 2.78 1.03 consensus index. Furthermore, the standard error of the not like mathematics. data measured using the improved Likert scale is less than Attitude towards 3.09 0.45 3.17 0.38 all others in all cases of sample size, so the sample mean is mathematics closer to the population mean. Most importantly, the data measured by the scale is normally distributed, indicating the inferential statistics are appropriate for the analysis. By using the method explained by Li (2013) to compare Thus, data measured using the improved Likert scale can be the quality of data, the samples of 100 students were set in applied for data analysis implementing descriptive statis- the research. Their attitude toward mathematics is tics. The data analysis is more appropriate than for the data measured by the ﬁrst question by using the Likert scale, the measured using the Likert scale. In addition, as the fuzzy Likert scale and the improved Likert scale. The improved Likert scale uses a measuring tool like the Likert improved Likert scale involved improvements based on the scale, it is more convenient to collect data by the improved Mamdani inference in four stages. The result from the Likert scale than with the scales proposed by Chang (1994), inference was 1.46, 2.24, 3.07, 4.05, and 4.55, respectively. Albaum (1997), and Hodge and Gillespie (2003). In partic- The answers of the samples are distributed in three cases. ular, it more convenient to collect data than using the fuzzy Statistical values of data were calculated and details are Likert scale proposed by Li (2013) because the fuzzy Likert provided in Tables 3e5. scale is appropriate only for speciﬁc topics, where there is From Tables 3e5, it was found that the arithmetic usually some quantitative data obtained from respondents means of data as measured by the Likert scale and the fuzzy used to assign the membership value for their answer Likert scale were equal to 3 in all cases, which shows that which is slightly complicated. However, constructing and the arithmetic mean determined using the two scales did improving the Likert scale with fuzzy logic may cause
P. Vonglao / Kasetsart Journal of Social Sciences 38 (2017) 337e344 343 Table 4 Statistics according to the distribution of data as measured by the fuzzy Likert scale Case Fuzzy Likert scale X S.D. Cns 1 1.5 2 2.5 3 3.5 4 4.5 5 1 0 0 0 30 40 30 0 0 0 3 0.39 .88 2 25 25 0 0 0 0 0 25 25 3 1.78 .16 3 10 10 10 10 20 10 10 10 10 3 1.23 .54 Table 5 Statistics according to the distribution of data as measured by the an ordinal level and it is not appropriate to analyze the data improved Likert scale using the arithmetic mean and standard deviation or to Case Improved Likert scale X S.D. Cns apply any inferential statistical methods. The statistical 1.46 2.24 3.07 4.05 4.55 method often applied to analyze data measured using a Likert scale merely depends on the assumption of Likert 1 0 0 100 0 0 3.10 0.00 1 (1932) that the data is on an interval level. The current 2 50 0 0 0 50 3.01 1.56 .00 3 15 15 40 15 15 3.07 0.99 .55 research successfully transferred the Likert scale to a suit- able scale by using fuzzy logic. This research found that the Likert scale could be improved by applying the fuzzy inference of Mamdadi which consisted of four important difﬁculties when adjusting the scale in the fuzzy inference steps: (1) fuzziﬁcation, (2) fuzzy rule evaluation, (3) ag- process. The validity of measurement depends greatly on gregation, and (4) defuzziﬁcation. Furthermore, a compar- key factors such as an appropriate membership function ison of data quality showed that the data measured using and suitable fuzzy rules. These factors mainly depend on the improved Likert scale with fuzzy logic was more suit- the expert's discretion. able to be analyzed with the arithmetic mean and standard deviation than data measured by the Likert scale. Impor- Conclusion and Recommendation tantly, the data were normally distributed and the standard error was lower. Therefore, it was appropriate to analyze Although the Likert scale had been widely used to the data by using statistical inference. For these reasons, measure latent variables, data content from the scale is on researchers should undertake data collection by latent Table 6 Normal distribution testing using KolmogoroveSmirnov test Scale m s Absolute Positive Negative z p Likert scale 3.098 0.452 0.08 0.079 0.08 1.395* .041 Improved Likert scale 3.168 0.379 0.058 0.037 0.058 1.012 .257 *p < .05. Table 7 Sample size, mean, standard error and standard deviation for different Likert scale approaches n Likert scale (m ¼ 3.098) Improved Likert scale (m ¼ 3.168) X S.E. S.D. X S.E. S.D. 30 3.1556 0.07928 0.43425 3.2111 0.06712 0.36762 40 3.1167 0.07716 0.48803 3.1754 0.06237 0.39445 50 3.1750 0.06659 0.47088 3.2290 0.05561 0.39320 60 3.0806 0.05176 0.40097 3.1507 0.04347 0.33674 70 3.0250 0.06411 0.53639 3.1057 0.05362 0.44859 80 3.1542 0.05880 0.52589 3.2105 0.04932 0.44110 90 3.1185 0.04291 0.40710 3.1864 0.03663 0.34751 100 3.0867 0.04639 0.46389 3.1534 0.03840 0.38398 110 3.1136 0.04326 0.45374 3.1811 0.03616 0.37923 120 3.0785 0.04567 0.50027 3.1478 0.03791 0.41529 130 3.0929 0.04261 0.48587 3.1633 0.03590 0.40933 140 3.1339 0.03806 0.45034 3.1976 0.03181 0.37640 150 3.0972 0.03873 0.47440 3.1669 0.03224 0.39482 160 3.0656 0.03359 0.42486 3.1418 0.02844 0.35977 170 3.1015 0.03582 0.46698 3.1700 0.02998 0.39088 180 3.0917 0.03472 0.46578 3.1593 0.02904 0.38961 190 3.0825 0.03495 0.48181 3.1522 0.02926 0.40328 200 3.0712 0.03157 0.44642 3.1439 0.02653 0.37519
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